Theorem opelvvg 4773

Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
opelvvg	|- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( _V X. _V ) )

Proof of Theorem opelvvg

Step	Hyp	Ref	Expression
1		elex 2812	. 2 |- ( A e. V -> A e. _V )
2		elex 2812	. 2 |- ( B e. W -> B e. _V )
3		opelxpi 4755	. 2 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. e. ( _V X. _V ) )
4	1, 2, 3	syl2an 289	1 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( _V X. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   _Vcvv 2800   <.cop 3670   X. cxp 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4149  df-xp 4729

This theorem is referenced by:  relsnopg  4828  opvtxfv  15863  opiedgfv  15866